Potential weights and implicit causal designs in linear regression

When we interpret linear regression estimates as causal effects justified by quasi-experiments, what do we mean? This paper characterizes the necessary implications when researchers ascribe a design-based interpretation to a given regression. To do so, we define a notion of potential weights, which encode counterfactual decisions a given regression makes to unobserved potential outcomes. A plausible design-based interpretation for a regression estimand implies linear restrictions on the true distribution of treatment; the coefficients in these linear equations are exactly potential weights. Solving these linear restrictions leads to a set of implicit designs that necessarily include the true design if the regression were to admit a causal interpretation. These necessary implications lead to practical diagnostics that add transparency and robustness when design-based interpretation is invoked for a regression. They also lead to new theoretical insights: They serve as a framework that unifies and extends existing results, and they lead to new results for widely used but less understood specifications.